Harvesting Strategies for “Bianchetti” and “Blue Fish” in the Ligurian Sea (North Mediterranean)

Appl. Math. Inf. Sci. 10, No. 3, 861-874 (2016) 861 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.18576/amis/100305

Samrat Chatterjee1,2, Daniela Pessani3 and Ezio Venturino 1,∗ 1 Dipartimento di Matematica “Giuseppe Peano”, Universit`a di Torino, via Carlo Alberto 10, 10123 Torino, Italy 2 Translational Health Science and Technology Institute, NCR Biotech Science Cluster, 3rd Milestone, Faridabad – Gurgaon Expressway, PO Box #04, Faridabad – 121001, India 3 Dipartimento di Scienze della vita e Biologia dei sistemi, Universit`adi Torino, via Accademia Albertina 13, 10123 Torino, Italy

Received: 6 Oct. 2015, Revised: 6 Jan. 2016, Accepted: 7 Jan. 2016 Published online: 1 May 2016

Abstract: A three level trophic fish web model is proposed, consisting of clupeiforms pilchard (Sardina pilchardus) and anchovy (Engraulis encrasicolus), at the bottom, mackerel (Scomber scombrus) in the intermediate trophic level and tunny (Thunnus thynnus) as top predators. The model includes human harvesting. The clupeiforms population is further partitioned into three age classes, eggs, larvae and the adult fishes to account for possible cannibalism but mainly for the fact that these larvae in the Mediterranean countries are highly appreciated as a delicacy for human consumption. In the absence of fisheries, to prevent total extinction and maintaining coexistence it is important that a sufficient amount of eggs fertilize to form larvae. The predation rate of larvae by other fish populations also has an important role, especially for the occurrence of periodic solutions. Different harvesting policies are then numerically simulated. Independently of the harvesting policy used, even a small amount of selective harvesting of larvae may cause total extinction of the system. To prevent it, fishing that avoids the catch of the larvae should be adopted. With non selective harvesting the system is preserved, but for certain parameter ranges oscillations arise, which under unfavorable environmental perturbations may lead the food chain to collapse.

Keywords: Trophic levels; Food web; Cannibalism; Extinction; Harvesting strategy; Stage-structured model

1 Introduction In this investigation we consider a complex situation in which several fish populations interact in a food web, to which is added also the external human intervention, Starting from the seventies, researchers in mathematical represented by fishing. We mainly concentrate here on the biology have devoted much effort to the study of food fundamental role that younglings have in shaping the chain systems [17,25,30]. Also models for cannibalism population dynamics and how this ultimately affects the have been investigated, since they frequently occur in future outcome of the whole aquatic food chain. This is natural systems, [2,23,38]. More recently, in the past particularly important in a closed sea such as the twenty years, several models for plankton dynamics have Mediterranean. In part larvae can be subject of been formulated, [9,19,33,34]. This is a relevant research cannibalism by their own adults, [28], and above all they topic because of its importance for human feeding. In fact are harvested by man. Both these aspects are suitably plankton lies at the bottom of the food chain in the ocean, taken into account in the model. The aim is the and thus ultimately also affects fisheries. Some of the investigation of selective harvesting on the larva of a models combined also the physical features of the ocean, specific fish population, in presence of its predator and a [31], other recent research efforts have been dedicated to top predator. We compare the results with unselective the issue of pattern formation, [16,27,24], and the fishing on all the fish populations. occurrence of red, or brown, tides, [6,7], which also have In Italy the whitebait of anchovy and pilchard are a negative effect on important human industries such as indicated with the vernacular term of bianchetti (or fisheries and tourism. gianchetti). As in use for other fish species both in Europe

∗ Corresponding author e-mail: [email protected]

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and in other countries of the world, such as New Zealand, and tunny, in turn gather into schools with the aim of both Australia, China, in Liguria and in other Italian Regions increasing their field of vision, therefore the hunting area, also the whitebait of anchovy and pilchard remains in the and of hunting in a coordinate and cooperative way. For human alimentary tradition as a delicacy and expensive instance tunnies form sets of 10-15 individuals which food. In fact, while fishing of bianchetti of anchovy is arrange themselves in a parabolic formation, with the forbidden according to rules of the European Community concavity ahead. In this way, the prey are gradually (CE 2550/2000), fishing of pilchard bianchetti is still surrounded and the predators save energy. Moreover, allowed, though regulated by norms and periods which among the reasons leading to the formation of wide fish vary every year and depending on geographic locations: gatherings, there could be a number of advantages, among usually it is performed for 60 days in the period between which energy saving, as mentioned, and also more ease in January and March [12]. Specifically, in 2010 the period reproduction [5]. has been from February 15th until April 15th. Furthermore, blue fishes include migratory species, Considering the size of the young fishes, from 7,000 to moving in the water both vertically and horizontally; the 12,000 individuals can be found in 1 kg of catch and reasons, the duration, and the extent of the migrations during a season of fishing, along the Ligurian coast, the vary from species to species but are mostly connected authorized boats can fish up to a few tons of bianchetti. with trophism and reproduction, in addition to Recently, consumption of the blue fish has increased, even environmental reasons, such as salinity and temperature. if the rate of catch is flatly decreasing. Recent reports, In the Mediterranean, the species belonging to blue [14], indicate that also bianchetti consumption in Italy has fish have size ranging from 10 cm up to 1 m; in addition increased in the past years, despite a decline of the catch. to the clupeiforms pilchard (Sardina pilchardus) and The paper is organized as follows. After giving some anchovy (Engraulis encrasicolus), they include different biological background, in Section 3 we describe the species, such as mackerel (Scomber scombrus) and tunny model. In Section 4 we analyze the system with no (Thunnus thynnus). harvesting, establishing the equilibria and their stability, The anchovy makes remarkable migrations analytically or by means of simulations. In Section 5 approaching the coast in the spring while going down, in harvesting is taken in consideration, with a comparison of winter, to depths of more than 100 m; it has trophic several options for the strategies. A final discussion diurnal habits and feeds on zooplankton, selecting prey concludes the paper. one by one. The reproduction occurs from April until September, with a peak in June-July; its maximum length is about 20 cm, [4]. The pilchard makes vertical and 2 Biological background horizontal migrations according to the temperature. It feeds on plankton filtered by means of branchial spines, The term “blue fish” identifies a group of deep-sea thin and dense. The reproduction occurs all year round species, characterized by a dark blue color, also with and shows a maximum in winter; the maximum size is green shades, on the back and on part of both sides as about 20 cm [4]. At the hatch the bianchetti are 2 mm well as a silver color on the sides and the abdomen. In long and maintain a transparent body till they reach the water, this color takes on a mimetic effect, since the fish length of 35-40 mm: at this point, the body begins to becomes invisible for predators, both fishes and pigment and the pigmentation is complete when the size cetaceans, which prey on it from the bottom upwards, as reaches 60 mm (“dressed” bianchetti). The bianchetti live well as horizontally, or from the top, mainly the birds. near the coastline and move to the open sea only when The blue fish are mostly gregarious, the gathering in form they have reached maturity [5]. of school reducing for the single specimen the possibility The mackerel makes seasonal migrations, being in of being predated. This occurs because the predator, in deep waters during the winter and moving towards the front of a remarkable number of prey, feels difficult to coast during the reproductive period, i.e. around the end select the individual fish to attack, [20,26,35]. Already of the winter to the beginning of the spring. Since the almost a century ago it was observed that schools of beginning of autumn, it feeds mainly on small sardines can confuse Great Northern loons, [1]. In other clupeiforms. The maximum size of the species is 30-35 situations the same phenomenon is observed, for instance cm, exceptionally up to 50 cm, and the maximum weight Japanese honeybees form a defensive ball around is about 1.5 kg [3]. attacking hornets, [22], while flocks of Bush tits detecting The tunny ravenously feeds on all the fishes it meets; hawks make a confusion chorus, [18]. In fact, considering the fish interrupts feeding only during the reproductive the limited visibility in water, every fish can be seen by a period, i.e. around May and June, when it moves towards predator in a restricted field, defined by the maximum the coast; it begins again the trophic migration to the open distance of visibility: if the fishes of the school swim very sea at the end of summer or beginning of the autumn. The near each other, they cause an overlapping of fields and tunny length is well over 1 m and up to 3 m, with a the probability by the predator to sight a school on the weight from 30-40 kg up to 400 kg [3]. open sea becomes only slightly greater than that of In the Ligurian Sea (Northern Mediterranean), the sighting a single specimen. The predators, like mackerel waters 10 miles off the coast represent a “meeting point”

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for some species of blue fish, i.e. a trophic and migration respectively, µ is the loss rate of E, due to natural area [5]: anchovy and pilchard constitute prey for the mortality, dispersion due to the currents, predation by mackerel which in turn is prey for the tunny. When the other species; ν is again the loss rate of individuals in the mackerel is in shallower waters, at the end of winter, it L population, but in this case in addition to the above feeds also on bianchetti. Mainly when it is juvenile, the mortality factors, also the maturation process into adults tunny prey also the clupeiforms. Besides in the gut is accounted for; θ > e denotes the net loss in egg content of anchovy also the bianchetti have been found, population E due the cannibalism of the adult fish S on its [28], evidently selected by the anchovy during its feeding own eggs and the fecundation process for which eggs “one by one”. During its non selective filter-feeding on mature to the larval stage. The parameter m is the planktonic organisms, the pilchard eats its own eggs and reproduction rate of the S population; the mackerels F small larvae. hunt at rate a1 on larvae L and a2 on pilchards S with The man feeds on all the levels of this trophic chain: mass conversion factors k1 and k2 respectively; similarly in the Ligurian Sea the blue fish represents the b and h respectively denote the predation rates and predominant part of the superficial fishing, performed by conversion factors of P on S while c and g are again the means of seines, having mesh-size and length which vary predation rate and conversion factor of tunnies P by according to the fish species to be captured. Lately the hunting F’s. The natural death rates of S, F and P are recommendations relevant to correct human feeding respectively n, f and p. induce an increased consumption of the blue fish, whose The first, second and the third equations denote the meat is rich in unsaturated fatty acids. evolution of the various life stages of the prey S. In the first equation for the eggs dynamics, the first term denotes the loss of eggs due to natural mortality, predation by 3 Model formulation other species and dispersion. The second term represents new recruits due to reproduction of the adults. Their We begin by illustrating the system characteristics. There environment is a large area with enough resources, so that are two adult populations involved, the top, P, and Malthus growth can be assumed. The last term models intermediate, F, predators, tuna and mackerel cannibalism by S and maturation to become bianchetti respectively, while in view of the interest in the fishing of due to hatching. Note that both these processes are related bianchetti we partition anchovies and pilchard, among to encounters of individuals in the two populations either adults, larvae and eggs, denoted by S, L and E for predation or for mating, as eggs are fertilized in water. respectively. For mathematical simplicity, we at first The second equation shows the larvae dynamics. assumed the interactions among the populations to be of Individuals enter this class via maturation, first term, and the form Holling type I, i.e. governed by the mass action are subject to mortality or mature to become adults: in law. Later on, to observe the robustness of our results, we both cases their numbers decrease, and this is modeled by numerically tested the model also with a Holling type II the second term; finally they can be prey of the mackerels functional response. F. The pilchards S are modelled by the third equation. ν The model is They enter this class from matured larvae, at rate l < , are subject to the natural mortality, second term, and are dE = −µE + mS − θES preyed by mackerels and by tunnies. The mackerels dt dynamics benefits from hunting the larvae and the dL = eSE − νL − a LF − H (L) pilchards, first two terms, but is subject also to hunt by dt 1 1 tunnies and to its own natural mortality, third and fourth dS terms. The tunnies could in principle also feed on other = lL − nS − a SF − bSP− H (S) (1) dt 2 2 sources, but we disregard this situation here. They feed dF both on the pilchards and on the mackerels, first two = a k LF + a k SF − cFP − fF − H (F) dt 1 1 2 2 2 terms of the last equation, but die out exponentially in dP their absence, third term. = bhSP+ cgFP − pP − H (P). dt 2 Here, also human interference, i.e. fishing, is accounted for. The harvesting strategies for the larva and different 4 The system with no harvesting fish populations are modeled by the last terms of each equation, but for the first one, and respectively denoted by 4.1 Boundary equilibria and their stability H1 and H2. In this way we account for the fact that bianchetti may or may not be subject to the same fishing strategy as the other fishes, while all the adult fishes are Proposition 4.1. The system (1) with no harvesting, harvested with the same strategy. H1 ≡ H2 ≡ 0, has the following boundary equilibria in the The parameters e and l represent the maturation rates E − L − S − F − P phase space, in addition to the trivial from egg to larva and from larva to adult population equilibrium in which each population vanishes,

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V0 ≡ (0,0,0,0,0): other types of food. At the peak of the summer instead, after the reproduction season, the feeding of the young νn n∆ µνn V1 ≡ , ,∆,0,0 , ∆ = , mackerel is rather voracious and based on everything they el l mel − θνn can find, whether be it pilchards’ eggs, pelagic peΘ p leΘ − nν mp crustaceans, clupeiforms or zooplankton. V2 ≡ Θ, , ,0, , Θ = . bhν bh νb µbh + θ p Proposition 4.2. Exploring the case in which mackerel divert their attention from bianchetti, i.e. setting a = 0, There is another boundary equilibrium point 1 the equilibrium point V3 can be explicitly evaluated, V3 ≡ (E3, L3, S3, F3, 0) with P = 0, where m f f (n + a2Ω) f mS3 S3(n + a2F3) V3 ≡ , , , Ω, 0 , E = , L = , a k µ + θ f a k l a k 3 µ + θS 3 l 2 2 2 2 2 2 3 b f l where, S3 = a1k1(n + a2F3)+ a2k2l eflm − νµa2k − νnµa k − νnθ f − νa θ f Ω = 2 2 2 2 2 . where F3 is a root of the equation νθ f + νµa2k2 4 3 2 Q1F + Q2F + Q3F + Q4F + Q5 = 0 (2) Remark 4.2. In this case, i.e. for a1 = 0, V3 is feasible for 3 2 2µ > νµ 2 ν µ ν θ ν θ with Q1 = a1k1a2 and eflm a2k2 + n a2k2 + n f + a2 f . 2µ νµ Proposition 4.3. Next if we assume that F does not feed Q2 = (a1k1n + a2k2l)a1 k1a2 + a1k1a2( a1k1a2 on S, i.e., a2 = 0 the equation (2) becomes, a2µk n a µa k l a θ f l a2µk a2 + 1 1 + 1 2 2 + 1 + 1 1 2 2µ θ 2 νθ θ 2µ 2 2 (a1 k1n + a1 f l)F + ( f l + a1n f l + a1 k1n +a a2µk1n), 1 +νµa k n)F − e f l2m + νnθ f l + νµa k n2 = 0. (3) νµ 2µ µ 1 1 1 1 Q3 = (a1k1n + a2k2l)( a1k1a2 + a1 k1n + a1 a2k2l 2 2 2 The equation (3) has a unique positive real root if and +a1θ f l + a µk1a + a a2µk1n)+ a1k1a2(a1nθ f l 2 2 1 2 1 only if e f l m > νnθ f l + νµa1k1n . Thus when a2 = 0, 2 µ νµ µ νθ +a1a2 k1n + a2k2l + a1n a2k2l + f l the equilibrium point V3 is feasible for +a2µk n2 + νa µa k n + a µa2k l 1 1 2 1 1 1 2 2 e f l2m > νnθ f l + νµa k n2. (4) νµ 2 νµ θ 1 1 + a1k1a2 + a1k1n + a1a2 f l), 2 The stability properties of these equilibrium points are Q4 = (a1k1n + a2k2l)(a1nθ f l + a a2µk1n + νµa2k2l 1 studied through the Jacobian matrix analysis. The Jacobian µ 2µ 2 νθ ν µ +a1n a2k2l + a1 k1n + f l + a2 a1k1n J at an arbitrary point is given by the matrix below with µ 2 νµ 2 νµ θ +a1 a2k2l + a1k1a2 + a1k1n + a1a2 f l) J44 = a1k1L + a2k2S − cP− f , J55 = bhS + cgF − p, ν µ ν µ νµ 2 +a1k1a2( n a2k2l + a2 a1k1n + a1k1n −µ − θS 0 m − θE 0 0 ν θ νµ 2 2 ν θ ν + n f l + a2k2l − e f l m + a2 f l),  eS − − a1F eE −a1L 0  Q = (a k n + a k l)(νµa2k l − e f l2m + νnµa k l 0 l −n − bP− a2F −a2S −bS . 5 1 1 2 2 2 2 2 2  0 a k F a k F J −cF  ν µ ν θ ν θ νµ 2  1 1 2 2 44  + a2 a1k1n + n f l + a2 f l + a1k1n ).  0 0 bhP cgP J   55  (5) Proof. Since Qi s’, i = 1,2,3 are positive, equation (2) Proposition 4.4. V0 is always stable. has a unique positive root if and only if Q5 < 0. This is only possible if e or m or both are very large so that the Proof. The eigenvalues associated with the matrix (5) term e f l2m becomes greater than the other positive terms evaluated at the origin are −µ, −ν, −n, − f , −p. Since present in the expression of Q4 and Q5. Hence, the all the eigenvalues are negative real numbers, the claim necessary and sufficient condition for the feasibility of a follows. unique boundary equilibrium point V3 is that em should Proposition 4.5. V1 is always unstable. exceed a threshold value. Further, from the analysis of V 3 Proof. Two eigenvalues associated with the matrix (5) at we need either a =6 0 or a =6 0 for the feasibility of S , 1 2 3 V are explicitly given by a k n∆l−1 +a k ∆ − f , bh∆ − p i.e., F has to feed on either L or S. This is obvious, since 1 1 1 2 2 and the remaining ones are the roots of the cubic equation no other food sources are present for the mackerels, and in their absence the latter would disappear. 3 2 Y + ω1Y + ω2Y + ω3 = 0, (6) Remark 4.1. The feasibility condition for V1 is given by mel > θνn, for V2 feasibility is ensured by leΘ > nν. where ω1 = n+ν + µ +θ∆, ω2 = (µ +θ∆)(n+ν), ω3 = Generally it is the size of the predator that selects the ∆(elm − θνn). From the existence criteria we have mel > size of the prey. In the autumn, rather than pilchards’ eggs nνθ, so ω3 < 0. Hence, by the Routh-Hurwitz criterion the or zooplankton, mackerel prefer, or find more abundant, claim follows.

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Proposition 4.6. V2 is stable if and only if the following 4.2 The coexistence equilibrium conditions hold: η4 > 0, η2 > η3η1η4 and a1k1 pmeΘ + a2k2 pν < cmlehΘ + cnhν + fbhν. Proof. The eigenvalues of the Jacobian (5) at V2 are Θ ν Θ ν ν We now turn to simulations. Fecundity of pilchards a1k1 pe + a2k2 p − cleh − cnh − fbh depends on geographical area, ranging from 50000 to bhν 60000 and from 76000 to 490000 eggs, with spawning all and the roots of the quartic equation year round once or twice per year. The larval stage duration from egg to adult lasts 40 days, [11,37]. Based 4 3 2 η1Z + η2Z + η3Z + η4Z + η5 = 0, (7) on this information we consider two possible scenarios, one with low reproduction rate and one with a higher one. with coefficients In the first case, assuming 72000 eggs per year mature, we get m = 200 per day, while in the latter one we have η = νbh, 1 taken 360000 eggs per year to mature, giving m = 1000 η Θ ν2 µν θ ν 2 = le bh + bh + bh + p , per day. Splitting evenly the maturing duration among egg η3 = pbhleΘ − pbhνn + leΘµbh + leΘθ p to larvae and larvae to adult, we take e = 0.05 and ν +ν2µbh + ν2θ p, = l = 0.05. For the mortality, [10], the natural mortality amounts to 0.64 per year, slightly lower, 0.5, in the η 2 Θθ ν2 µν 2θν ν 4 = p le − p nbh − p nbh − p n − plem Adriatic, [29], giving n = 0.002 per day. For Tunnus +leΘθ pν + pleΘνbh + pleΘµbh, thynnus natural mortality is 0.14 per year for all ages, 2 2 2 2 [15], giving p = 0.0004 per day. We use the same values η5 = pνleΘµbh + p νleΘθ − pν µnbh − p ν θn. also for mackerel, f = 0.0004. Here η1 and η2 are always positive. From the existence criteria of V2, η3 and η5 are also positive. The Routh Hurwitz conditions indicate that all the roots of the equation (7) have negative real parts if and only if Table 1: The parameter values: those that are found in the η1 > 0, η3 > 0, η4 > 0 and η2 > η3η1η4. literature, [11,37,10,29], are reported with a star, the remaining Since it was not possible to find a closed form for V3, it is not possible to find the stability conditions explicitly for ones are hypothetical. Name values Name values V and hence this analysis is omitted. However, V takes a 3 3 *m 200 day−1 a 0.1 kg−1day−1 simplest form with a = 0 and so the stability property at 2 1 g 0.01 *ν 0.05 day−1 V is studied with a = 0. b 3 1 µ 0.3 day−1 b 0.0015 kg−1day−1 Proposition 4.7. The equilibrium point V (assuming a −1 −1 −1 −1 b 3 1 = *e 0.05 kg day c 0.27 kg day 0) is stable if and only if both the following conditions hold *n 0.002 day−1 * f 0.0004 day−1 b −1 *l 0.05 day k1 0.6 bh f θ −1 −1 + cgΩ < p, ζ4 > 0, ζ2 > ζ1ζ3ζ4. 0.2 kg day k2 0.5 a k −1 −1 −1 2 2 a1 2.3 kg day *p 0.0004 day h 0.0001 – – Proof. The eigenvalues of the Jacobian (5) at V3 with a1 = 0 are bh f b + cgΩ − p, a2k2 The set of parameter values taken from literature are and the roots of the quartic equation, summarized in Table 1. For further analysis the interior 4 3 2 equilibrium is investigated by means of numerical ζ V + ζ V + ζ V + ζ V + ζ = 0 (8) 1 2 3 4 5 simulations using the Matlab built-in routine ode45 with 2 where ζ1 = a2k2, ζ2 = νa2k2 + na2k2 + a Ωk2 + ρ1, the parameter values given in Table 1. Under these 2 conditions, the coexistence equilibrium is stable, see ζ 2Ω ν 2 Ων 3 = a2 fk2 + na2k2 + a2k2 Figure 1. ρ Ω ν lemfa2k2 Taking now the conversion rate from egg to larva as a + 1(n + a2 + ) − ρ , 1 parameter to study, we find that when e is reduced to ζ 2 Ων ρ Ω ν Ων . 4 = a2k2 f + 1(a2 f + n + a2 ) 0 000071, all populations vanish and the system collapses, see Figure 2. Thus total disappearance is lemfa2k2µ −lemf − ρ , possible if a large enough amount of eggs does not hatch 1 to become larvae, supporting and furthering our analytical ζ5 = fa2Ωρ1ν and ρ1 = µa2k2 + θ f . results of the former Subsection on the local stability of Since ζ2 > 0 and ζ5 > 0, the claim follows. the origin.

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indiscriminately on all the web populations, respectively. 1000 1000 The case of ﬁshing on larvae only is also contemplated although less realistic, as the adult ﬁsh catching in general L E 500 500 continues the whole year round.

0 0 0 5000 10000 15000 0 5000 10000 15000 Time Time

10 5.1 Linear harvesting policies 2 F S 5 1 We study the system with the assumption that the harvesting H1 is done on the larvae L via the function 0 0 0 5000 10000 15000 0 5000 10000 15000 H1 = q1L, q1 being the harvesting rate of the larvae, while Time Time the adult ﬁsh populations S, F and P are harvested at a 4000 different rate q2, via the functions H2(S) = q2S, 3000 H2(F)= q2F, H2(P)= q2P.

P 2000 With these assumption on H1 and H2 the system (1) 1000 again has three boundary equilibrium points in addition to

0 the origin. These equilibria are 0 5000 10000 15000 Time (ν + q1)(n + q2) (n + q2)Φ U1 ≡ , ,Φ,0,0 , el l Fig. 1: For the parameters given in Table 1, the system (1) µ(ν + q )(n + q ) without harvesting shows a stable interior equilibrium steady Φ 1 2 = θ ν , state. mel − ( + q1)(n + q2) (p + q2)eΨ (p + q2) leΨ n + q2 U2 ≡ Ψ, , ,0, − , bh(ν + q1) bh (ν + q1)b b m(p + q2) 400 1 Ψ = , µbh + θ(p + q2) L E 200 0.5 U1 and U2 are feasible respectively for

0 0 0 5000 10000 15000 0 5000 10000 15000 mel > θ(n + q2)(ν + q1), leΨ > (n + q2)(ν + q1). Time Time 1 2 If we compare the values of the equilibria and their feasibility conditions for this case with those for the model F S 0.5 1 in absence of harvesting,we note that the only differenceis ν 0 0 in the scale, since here the constants , n, f , p of the model 0 5000 10000 15000 0 5000 10000 15000 with no harvesting get here replaced by ν + q1, n + q2, Time Time f + q2, p + q2. The same changes occur for the feasibility 1.5 of U3, similar to the one of V3, and for the stability of all 1 the equilibria. Thus we omit the analysis. P 0.5 From the available literature, [10], the ﬁshing mortality in the Adriatic, [29], is 0.30 per year so that q1 = 0.001 0 0 5000 10000 15000 per day. The ﬁshing mortality for Tunnus thynnus depends Time on the age, and we use an average value of q2 = 0.0004 per day. So, for further analysis we simulate the system Fig. 2: Total extinction of all the populations in system (1) with (1) numerically with the parameter values given in Table 1 no harvesting for a very low egg maturation rate e = 0.000071. with q1 = 0.001 and q2 = 0.0004 as found in the literature. In these conditions the populations coexist, see Figure 3. But if the harvesting rates q1 and q2, are both increased to the level 0.02, we observe total extinction of the system. 5 The various harvesting policies

Here we study the actual system (1) with different 5.1.1 Harvesting only of larvae harvesting policies, for the prey fish at the larval stage and for all the adult fish populations. This means that we take Although not much realistic, since fisheries operate two functional forms for H1 and H2 and combine them continually in time, we study here the food web with the also with selective and unselective harvesting, i.e. when assumption that the harvesting H1 is done only on the the latter is performed only on adult fish or larvae. Using again the set of parameter values given in

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1000 1000 400 10 L L E 500 500 E 200 5

0 0 0 0 0 5000 10000 15000 0 5000 10000 15000 0 50 100 150 200 250 0 50 100 150 200 250 Time Time Time Time 10 3 1.5 20

2 1 F F S 5 S 10 1 0.5

0 0 0 0 0 5000 10000 15000 0 5000 10000 15000 0 50 100 150 200 250 0 50 100 150 200 250 Time Time Time Time 4000 1.5

P 2000 P 0.5

0 0 0 5000 10000 15000 0 2000 4000 6000 8000 10000 12000 Time Time

Fig. 3: Coexistence for the system (1) is attained when the ﬁshing Fig. 4: Starting from the coexistence state of Figure 1 with efforts are q1 = 0.001 and q2 = 0.0004, as given in the literature. selective linear harvesting on L at rate q1 = 2.1, q2 = 0, the other Here the linear form of harvesting is used. The other parameters parameters being the same as in Figure 1, the food web collapses. are the same as in Figure 1.

400 2 Table 1, if we introduce fisheries with removal rate L E 200 1 q1 = 2.1, the whole system is driven to extinction, as depicted in Figure 4. Thus increase in harvesting on 0 0 larvae can drive the system to total extinction but that 0 500 1000 0 500 1000 need almost a 2000 fold increase than normal rate and Time Time 1 10 that too with the top predator taking much larger time F than the others species. Thus in case of linear harvesting S 0.5 5 strategy only on the larval population, it is very difficult 0 0 to cause total collapse of the food web. Recall however 0 500 1000 0 500 1000 that this situation is quite unrealistic, since adult fishes are Time Time caught generally the whole year round. 1.5 1 P 0.5 5.1.2 Harvesting only adult fish 0 0 500 1000 We consider the system (1) with the assumption that the Time harvesting is forbidden on the younglings, i.e. H1 = 0 and H2 is performed only on the adult fish populations S, F Fig. 5: Again for linear selective harvesting on adult fish only, and P. For small values of q1 the system is stable at the starting from the condition and parameter values of Figure 1, we coexistence steady state, but we do not report graphically obtain total extinction with fishing efforts q1 = 0, q2 = 0.01. this result here. As the value is increased to q2 = 0.01 we observed total collapse of the system, Figure 5. law of diminishing returns, [8,13,16,21], but see also [32]. The two functions for the larvae and for the adult 5.2 Harvesting with bounded maximal return fishes are respectively expressed by

We now consider a more realistic description of the ﬁsh q1L q2x catches, expressed by the Holling type-II function. For H1(L)= , H2(x)= . 1 + r1L 1 + r2x illustrating the features of this function with respect to the Holling type-I one can consult any standard text in The model (1) with these functions becomes too complex mathematical biology or even in operations research, to be studied analytically. We rather investigate it only by where the gain expressed by this function is known as the means of numerical simulations. For the usual set of

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parameter values of Table 1, and new parameter values 1.2 4 for the harvesting functions given by q1 = 0.001, q2 = 0.0004, r1 = 0.1 and r2 = 0.4, the system shows 1 3 L E 0.8 stable coexistence. The system remains unchanged for a 2 large range of r1 and r2. But, as expected, upon increasing 0.6 1 the values of qi’s, like in linear harvesting, here too the 0 50 100 150 200 0 50 100 150 200 system becomes totally extinct. Time Time 3 1

2 F S 0.5

6 A possibly endangered situation 1 0 0 50 100 150 200 0 50 100 150 200 We now consider a set of hypothetical values and perform Time Time bifurcation analysis to understand the possible ecological 2 danger associated with possible shifts of some parameter values due possibly to relevant exogenous changes in the P 1 environmental conditions. The hypothetical set of 0 parameter values, given in Table 2, are chosen so that the 0 50 100 150 200 system shows stable coexistence. We start by searching Time periodic solutions. Fig. 6: Coexistence of all the species via oscillations in system (1) with no harvesting when the F’s hunting rate a1 increases Table 2: A hypothetical set of parameter values. here from 0.65 to 0.75. Names values Names values −1 −1 −1 m 0.35 day a2 0.2 kg day g 0.6 ν 0.3 day−1 µ 0.3 day−1 b 0.6 kg−1day−1 e 0.8 kg−1day−1 c 0.7 kg−1day−1 n 0.14 day−1 f 0.4 day−1 −1 l 0.5 day k1 0.7 −1 −1 θ 0.2 kg day k2 0.2 −1 −1 −1 a1 0.65 kg day p 0.5 day h 0.5 – –

The values of different parameters are varied one by one, keeping all the other ones ﬁxed. It is observed that when the predation rate a1 on L by the predators F increases to 0.75, the system shows coexistence through periodic oscillations, see Figure 6. This is clearer from the bifurcation diagram obtained using AUTO, Figure 7, where we have taken the parameter a1 as the bifurcation parameter. It is observed that for low value of a1, the population F goes to extinction, while all the other ones attain a steady state, with constant values independent of Fig. 7: For the system (1) without harvesting, using AUTO, we the values of the parameter a1. Past the branch point in plot a bifurcation diagram taking the F’s hunting rate a1 as the terminology of AUTO, which in this case is rather a bifurcation parameter. At the critical point a1c there occurs a transcritical bifurcation point, all the populations coexist Hopf bifurcation. BP stands for branch point, i.e. a transcritical at a stable steady state with F increasing as a1 increases, bifurcation, and HB means Hopf-bifurcation. and all the other populations instead diminishing. When the value crosses some threshold value a1c, speciﬁcally here it is a1c = 0.67, at which a Hopf bifurcation occurs, the populations coexist through periodic oscillations. hunting rate of F’s on larvae therefore plays an important Thus we have proven by means of simulations the role in the dynamics of the whole system and existence of a critical threshold a1c where the Hopf consequently on the evolution of larvae and their bifurcation occurs around the positive steady state, predators. The same bifurcation is obtained for a2, i.e., thereby inducing oscillations of the populations. The predation rate on S by the predators F, not reported here.

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Fig. 8: Bifurcation diagram, obtained using AUTO, for selective Fig. 9: Bifurcation diagram, obtained using AUTO, as function adult fish linear harvesting, i.e. q1 = 0, taking q2 as bifurcation of the effort q1 for the Holling type II selective harvesting on parameter. Note the Hopf bifurcation at a lower critical value, larvae only. HB means Hopf-bifurcation. here the value is very small, and total extinction when a higher threshold is reached. BP stands for branch point, i.e. transcritical bifurcation, and HB means Hopf-bifurcation, SN is for saddle node bifurcation. steady state, which exists for a small range of q1, followed by a range of q1 for which all the populations coexist via limit cycles, past the Hopf bifurcation point located at q ≈ 0.12. In fact for this small value of q , a Interesting findings on bifurcation are observed in the 1 1 Hopf-bifurcation, HB2, occurs and sustained periodic model with harvesting, for example linear harvesting on solutions arise. adult fish only. To understand the changes in the dynamical behaviour of the species as functions of the Again to investigate the relation between the changes in the value of q2 we plot the corresponding capturing rate q1 and the saturation constant r1, we plot in bifurcation diagram in Figure 8. The structure of this the two dimensional r1 − q1 parameter space a bifurcation picture is rather complicated, showing a branch point, i.e. diagram, Figure 10. For low values of r1 and q1, we a transcritical bifurcation, two Hopf bifurcations and a observe a saddle-node bifurcation SN5. For very low saddle-node bifurcation. For a certain range of q2 all the values of r1 and q1 the system attains a stable coexisting populations coexist, then a bifurcation occurs and state, but by increasing either r1 or q1, the stability of the oscillations arise. In Figure 8 at the branch point, i.e. system is destroyed and the system shows periodic transcritical bifurcation, BP4, P vanishes while the other oscillations. populations remain in the system. As q2 increases past the We finally consider the harvesting with bounded branch point, i.e. transcritical bifurcation, BP4 there maximal return for all the adult fishes, i.e. we set q1 = 0. occurs a Hopf-bifurcation, HB5, among the remaining With the parameter values given in Table 1 and fixing populations and then a saddle-node bifurcation, SN6, r2 = 1.2, we plot the bifurcation diagram, Figure 11, leading all the population to collapse, i.e. causing total taking q2 as the bifurcating parameter. The structure of extinction of the system. But already an important remark this diagram is again rather complicated, showing a Hopf here is that even for a not so large value of the harvesting bifurcation HB2 for very low values of q2, around 0.04, rate, q2 ≈ 0.17, the top predator population disappears. leading to an unstable manifold for which the system Next we consider the model in the presence of populations start all to oscillate giving rise to limit cycles harvesting with bounded maximal return. We consider in the phase space. With an increase in q2, we observe a this type of harvesting only on the larvae, i.e., we have saddle-node bifurcation (SN6) and a branch point, i.e. system (1) with q2 = 0 ≡ H2. With the parameter values transcritical bifurcation, located at q2 ≈ 0.22. Past this given in Table 1 and r1 = 0.7, the bifurcation diagram is value, it is seen from the figure that the top predator plotted in Figure 9 taking q1 as the bifurcating parameter. population gets extinguished. But the presence of the The changes in the dynamical behaviour of the species as saddle-node bifurcation itself ultimately drives the whole functions of the larvae capturing rates show an initial system to extinction.

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Fig. 10: Two parameter space diagram, obtained using AUTO, Fig. 12: Two parameter space diagram, obtained using AUTO, showing the stability regions as functions of q1 and r1 for the showing the stability regions as functions of q2 and r2 for the Holling type II selective harvesting on larvae only. A saddle-node Holling type II selective harvesting only on adult ﬁshes. Two (SN) bifurcation is observed for low values of q1 and r1. CUSP bifurcation are observed, both for low values of q2, but one for low value of r2 and the other one for a high value of r2.

We also provide a two-parameter space diagram in terms of q2 and r2. In addition to the saddle node bifurcation, located on the bottom left and almost overlapping a cusp bifurcation, we observe the presence of a second cusp bifurcation for a moderate value of q2, see Figure 12. The cusp bifurcation implies the presence of a hysteresis phenomenon. For the ﬁrst one of the two, this is also clear e.g. from Figure 11 looking at the behavior of the top predator population P near q2 = 0.22.

6.1 Numerical Result with Holling type-II interaction To understand the effect of predator saturation, we now perform a numerical test on the model (1) modiﬁed with Holling type-II functional response as the interaction term and then compare the result with the original model (1). With this assumption, the model (1) becomes Fig. 11: Bifurcation diagram, obtained using AUTO, as function dE of the effort q2 for the Holling type II selective harvesting only = −µE + mS − θES on adult ﬁshes. BP stands for branch point, i.e. transcritical dt bifurcation, and HB means Hopf-bifurcation, SN is for saddle dL ν a1LF node bifurcation. = eSE − L − − H1(L) dt 1 + β1F dS a2SF bSP = lL − nS − − − H2(S) (9) dt 1 + β2F 1 + γ1P dF a1k1LF a2k2SF cFP = + − − fF − H2(F) dt 1 + β1F 1 + β2F 1 + γ2P dP bhSP cgFP = + − pP− H2(P). dt 1 + γ1P 1 + γ2P

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Fig. 13: For the system (9) without harvesting, using AUTO, Fig. 14: We plot a bifurcation diagram for the system (9) with a1 we plot a bifurcation diagram taking the F’s hunting rate a1 as as bifurcation parameter for a higher predator saturation constant bifurcation parameter. Here also, like in case of system (1), there β1 = 0.2. In this case no Hopf bifurcation occurs. is a critical point a1c where Hopf bifurcation occurs.

To capture the robustness of the results we start with the parameter values corresponding to Table 2, with β1 = 0.1, β2 = 0.2, γ1 = 0.16, γ2 = 0.2. The system (9) without harvesting shows a stable interior equilibrium steady state. Next we plot the bifurcation diagram with the hunting rate a1 as bifurcation parameter and observe the same qualitative result, see Figure 13. But when we increase β1 to 0.2, the effect of predator saturation is immediately observed, with no Hopf-bifurcation occurring in the bifurcation diagram, see Figure 14.

To observe the effect of harvesting we first simulate the system (9) with linear harvesting with q1 = 0.07 and q2 = 0.06 keeping the other parameters fixed and observe periodic oscillations. We again observe periodic solutions Fig. 15: Bifurcation diagram for the system (9) as function of when we simulate the system (9) with Holling type II the effort q2 for the Holling type II selective harvesting only indiscriminate harvesting where q1 = 0.07, q2 = 0.06, on adult fishes. BP stands for branch point, i.e. transcritical r1 = 0.6, r2 = 0.9. These results are in agreement with the bifurcation, and HB means Hopf-bifurcation, SN is for saddle results we obtained for the model (1) with a difference in node bifurcation. the quantitative value of q1 and q2. Finally we perform a bifurcation diagram taking q2 as the bifurcation parameter, see Figure 15. The structure here is the same and as complicated as we observe in case of the linear interaction term. Thus we may conclude that the result we obtain from the linear interaction term model (1) is robust enough to hold true for other models with different functional responses.

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7 Discussion total extinction, this kind of selective harvesting must be avoided to prevent extinction of the food web. Here we considered a five dimensional population model, With the hypothetical less realistic set of parameter modelling the situation in the Ligurian sea. This for two values for endangered ecological situation, we observed main reasons. The Ligurian sea is one of the most that for too low value of the hunting rate a1 of the important places in the Mediterranean where the catch of mackerels on larvae, the former become extinct. For other bianchetti is exerted. Secondly, this catch is not present in situations in nature leading to such outcome, see [36]. all the Mediterranean, since in other cultures perhaps the This is understandable since larvae and pilchards in this larvae are less palatable, or due to the local environmental model are the only food source of the mackerels F and conditions, bianchetti are seldom present. The food web the combination of a low a1 with a low hunting rate a2 of accounts for the fact that clupeiforms constitute the prey the mackerels on pilchards S (Table 2) causes the for mackerels and both are predated by tunnies. The extinction of mackerels. For a moderate value of a1, all model is set up also to investigate how the possible the populations coexist at a stable steady state, bur if the cannibalism of clupeiforms on their larval offsprings and value of this removal rate further increases and crosses a the latter being subject to capture by selective fishing to threshold value a1c, a Hopf bifurcation occurs and the get the valuable bianchetti affects the long term system populations now coexist by exhibiting persistent periodic dynamics. In the absence of this selective harvesting, the oscillations. These remarks enlighten the role of the predators-free equilibrium is never stable, so that removal rate a1 in shaping the dynamics of the food clupeiforms with their eggs and the bianchetti never settle chain. to constant values. The conditional stability of the other Under the same ecological endangered situation, two boundary equilibria shows instead that one of the when harvesting only on adult fish, the saddle-node intermediate predators may disappear, under certain bifurcation occurs for values of q and r that are larger if conditions on the model parameters. This is certainly not 2 2 compared with the parameter values q1 and r1 at which it a good implication, since it negatively affects the fishing occurs when only larvae are harvested. Thus the chance industry. of extinction is higher for the latter case than in the Adding also a linear fishing effort in the system, the former. In other words, selective harvesting affecting only few analytical results indicate that the model retains the the adult fishes has a lower environmental impact on the same qualitative behavior as in its the absence, with only whole food chain compared to selective harvesting quantitative differences, depending on the value of the performed only on larvae. The chance of collapse of all harvesting rates qi, i = 1, 2. the populations is smaller, although the coexistence We consider a set of parameter value representing the equilibrium may become unstable and limit cycles around real world situation, taken from different literature. In the it may well arise for smaller value of q1, see the Hopf absence of harvesting, numerical experiments exhibit bifurcation point HB2 in Figure 11. total extinction if a large enough amount of eggs does not hatch to become larva, in agreement with the analytical To assess the robustness of the model, we performed result. The same result is observed for the model with numerical simulations for the same model in which the high harvesting rates. In the presence of linear harvesting Holling type II functional response is used. The result affecting both the larvae and the fishes, the system may obtained for the original model holds true also for the go toward extinction even for small values of the Holling type II functional response when the predator harvesting rates of bianchetti. With selective fishing saturation is kept below a certain value. For higher values performed only on the larvae, there is a high chance of of predator saturation, the system never reaches periodic total extinction. Harvesting only on larvae is however a coexistence, rather switches between stable coexistence rather unrealistic situation, as fisheries operate and extinction. continually and therefore the harvest on the adult fish The fishing of the different species of both the blue populations in reality never ceases. But such result should fish, in general, and bianchetti, in particular, can thus warn us of its possible undesirable consequences. affect the various links of the natural trophic chain, with We also considered the Holling type-II form of consequences which could become remarkable or actually harvesting, which is more realistic as the Ligurian sea is irreparable. In spite of its economic palatability, then it rather small, so that for increasing fishing efforts and seems that fishing of bianchetti should be avoided to limited resources, diminishing returns are expected. preserve the long term survival of the food chain resource. Assuming to hunt all the populations in the web, the same phenomenon occurring for the linear harvesting arises in this case as well. Changes are here observed only in the values of the fishing rates for which the system Acknowledgement behaves differently, i.e. there are quantitative but not qualitative differences with the previous situations. Therefore, since selective harvesting only on larvae, Research supported by: MIUR Bando per borse a favore di independently of the specific harvesting policy, may cause giovani ricercatori indiani.

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References heterogeneity: a model study, Math. Model. Nat. Phenom. 3(3), 131–148 (2008). [1] W. E. Allen, Behavior of loon and sardines, Ecology 1, 309– [20] D.H. Morse, Feeding behavior and predator avoidance in 310 (1920). heterospecific groups, BioScience 27, 332–339 (1977). [2] F. Bekkal-Brikci, K. Boushaba, Ovide Arino, Nonlinear [21] J.D. Murray, Mathematical Biology, Springer Verlag, Berlin age structured model with cannibalism, Discrete and (1989). Continuous Dynamical Systems B 7, 201–218 (2007). [22] M. Ono, T. Igarashi, M. Sasaki, Unusual thermal defence [3] G. Bini, Atlante dei pesci delle coste italiane (Atlas of fish by a honeybee against mass attach by hornets, Nature 377, and Italian coasts), 6, Mondo Sommerso ed. (1968). 334–336 (1995). [4] G. Bini, Atlante dei pesci delle coste italiane (Atlas of fish [23] A. G. Payne, C. Smith, A. C. Campbell, A model of oxygen- and Italian coasts), 2, Mondo Sommerso ed. (1970). mediated filial cannibalism in fishes, Ecological Modelling [5] R. Cattaneo Vietti, L. Relini Orsi, M. Wurtz, La pesca 174, 253–266 (2004). in Liguria (Fishing in Liguria). Centro Studi Unioncamere [24] S. V. Petrovskii, H. Malchow, A minimal model of pattern Liguri (1985). formation in a prey-predator system, Math. Comp. Model. [6] J. Chattopadhyay, S. Chatterjee, E. Venturino, Patchy 29, 49–63 (1999). agglomeration as a transition from monospecies to recurrent [25] S. L. Pimm, J. H. Lawton, Number of trophic levels in plankton blooms, Journal of Theoretical Biology 253, 289– ecological communities, Nature 268, 329–331 (1977). 295 (2008). [26] T. J. Pitcher, J. K. Parish, Functions of shoaling behaviour in [7] S. Chakraborty, S. Chatterjee, E. Venturino, J. teleosts, in: Pitcher T.J. (Editor) Behaviour of teleost fishes, Chattopadhyay, Recurring plankton bloom dynamics Chapman and Hall, New York, 363–440 (1993). modeled via toxin-producing phytoplankton, J. Biological [27] J.-C. Poggiale, M. Gauduchon, P. Auger, Enrichment Physics 33, 271–290 (2007). paradox induced by spatial heterogeneity in a [8] C. Clark, Mathematical bioeconomics: the optimal phytoplankton-zooplankton system, Math. Model. Nat. management of renewable resources, Wiley, New York Phenom. 3(3), 87–102 (2008). (1976). [28] M. Romanelli, O. Giovanardi, A special fishery aimed at [9] A.M. Edwards, J. Brindley, Zooplankton mortality and the advanced larvae of Sardina pilchardus (Walbaum) along the dynamical behavior of plankton population models. Bull. northwestern and central western coasts of Italy: a general Math. Biol. 61, 202–339 (1999). report. Biologia Marina Mediterranea 7(3), 158–172 (2000). [10] Z. Erdogan, H. Torcu-Koç, S. GiciIi, G. Ulunehir, Age, [29] A. Santojanni, N. Cingolani, F. Donato, S. Colella, A. growth and mortality of European pilchard, Sardina Berlardinelli, G. Giannetti, I. Leonori, G. Sinovcic, B. pilchardus in Edremit Bay (northern Aegean Sea, Turkey), Marceta, Anchovy and sardine stock assessment in the GSA Cybium 34(2), 185–193 (2010). 17, 1975–2007, (2008), GFCM-SAC SubCommittee on [11] R. Froese, D. Pauly (Editors), Sardina Stock Assessment (SCSA), Izmir, 22-26 September 2008. pilchardus, in: FishBase (2006). [30] T.W. Schoener, Food webs from the small to the large, (http://www.fishbase.org/summary/Sardina- Ecology 70, 1559–1589 (1989). pilchardus.html). [31] L. Seuront, Microscale Complexity in the Ocean: [12] M. E. Gramitto, La gestione della pesca marittima in Italia Turbulence, Intermittency and Plankton Life, Math. (Marine fishing policies in Italy). Monografie scientifiche Model. Nat. Phenom. 3(3), 1–41 (2008). CNR (2001). [32] G. T. Skalski, J. F. Gilliam, Functional responses with [13] F. S. Hillier, G. J. Liebermann, Operations Research, Second predator interference: viable alternatives to the Holling type Edition, Holden-Day, (1974). II model. Ecology 82(11), 3083–3092 (2001). [14] Ismea, Compendio statistico del settore [33] I. Siekmann, H. Malchow, Local collapses in the Truscott- ittico (Summary of fish sector) (2009): Brindley model, Math. Model. Nat. Phenom. 3(4), 114–130 http://www.ismea.it/flex/cm/pages/ServeBLOB.php/L/IT/ (2008). IDPagina/4849 [34] J. Truscott, J. Brindley, Ocean plankton populations as [15] M. Labelle, T. Hoch, B. Liorzou, J. L. Bigot, Indices of excitable media, Bulletin of Mathematical Biology 56, 981– bluefin tuna (Thunnus thynnus thynnus) abundance derived 998 (1994). from sale records of French purse seine catches in the [35] G. Turner, T. Pitcher, Attack abatement: a model for group Mediterranean sea, Aquat. Living Resour. 10, 329–342 protection by combined avoidance and dilution, American (1997). Naturalist 128, 228–240 (1986). [16] H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal [36] C. Webb, A complete classification of Darwinian extinction Patterns in Ecology and Epidemiology – Theory, Models, in ecological interactions, The American Naturalist 161, and Simulation. Mathematical and Computational Biology 181–205 (2003). Series. Chapman & Hall / CRC, Boca Raton (2008). [37] P. J. P. Whitehead, G.J. Nelson, T. Wongratana, FAO Species [17] T. Matsuoka, H. Seno, Possibly Longest Food Chain: Catalogue. Vol. 7. Clupeoid fishes of the world (Suborder Analysis of a Mathematical Model, Math. Model. Nat. Clupeoidei). An annotated and illustrated catalogue of the Phenom. 3(4), 131–160 (2008). herrings, sardines, pilchards, sprats, shads, anchovies and [18] R. C. Miller, The significance of gregarious habit, Ecology wolf-herrings. FAO Fish. Synop. 125(7/2), 305–579. Rome: 3, 122–126 (1922). FAO (1988). [19] A. Morozov, E. Arashkevic, Patterns of zooplankton [38] A. Wikan, A. Eide, An analysis of a nonlinear stage- functional response in communities with vertical structured cannibalism model with application to the

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Northeast Arctic cod stock, Bull. Math. Biol. 66, 1685–1704 Ezio Venturino (2004). received the PhD in Applied Mathematics at SUNY Stony Brook. Afterwards he spent several years at the University Samrat Chatterjee of Iowa before returning received the PhD degree to Italy at the Polytechnic from the University of of Torino. Professor of Kolkata and spent two years Numerical Analysis at in a postdoctoral position the University of Torino since at the University of Torino. fifteen years, he authored He then joined ICGEB, New more than forty research papers in numerical analysis and Delhi as Research Scientist more than one hundredtwenty in mathematical biology, (Project) and worked there for the field in which presently lie his main scientific three years. Currently he is an interests. He also serves in the editorial board of Assistant Professor in THSTI, Faridabad, India. His several international Journals. Ha has acquired a large current research work involve cell biology, especially cell international experience visiting many scientific signalling. It involves theoretical approach using institutions in numerous countries worldwide. He mathematical models and systems biology to solve has actively taken part in more than hundredthirty biological problems in collaboration with experimental international conferences, organizing and chairing special biologists. He also works in field of ecology, both sessions. Recently he has been the main organizer of the terrestrial and marine. He wrote a book on migratory international MPDE14 conference that took place at the birds and spread of disease based on his PhD research University of Torino, August 25th-29th, 2014. published by VDM-Verlag (Germany). (ISBN-13: 978- 3639268256). He acts as reviewer of many journals and currently joined F1000 as Associate Faculty.

Daniela Pessani Associate Professor at the University of Torino, is the manager of the Zoology and Marine Biology Laboratory (LBMT), where six young researchers work in the area of marine biology. The topics of her personal research concern different aspects of marine biology: life history of species with planktonic developmental stages, conservation of endangered (both freshwater and marine) species, bioacoustics and welfare of penguins and dolphins. She has coordinated many field researches in the Mediterranean Sea and has taken part in a post-tsunami campaign in Thailand. Many scientific projects she presented have been financed by the Ministry of Environment, private Fundations and public organizations. She organized the international Congress on Decapod Crustacea (Turin, September 2008) and took part in many national and international congresses. She published about one hundred and fifty papers and some chapters of scientific books.

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